Continuous functions on path connected and connected sets, Real Analysis II
In this video, we prove two results that show how certain topological properties are preserved under continuous functions. (Review connectedness and path-connected here: • Connected and Path Connected Sets, Re... .)
First, we prove that if a continuous function f has a path-connected domain, then its image f(A) is also path-connected. We start by selecting two points in the image, trace them back to the domain, and use the fact that the domain is path-connected to construct a continuous path between them. Applying the function f to this path results in a continuous path in the image, proving that f(A) is path-connected.
Next, we prove that if f is continuous and the domain A is connected, then the image f(A) is connected. This proof takes a different approach using contrapositive. We assume that f(A) can be separated into two disjoint open sets and show that this would imply the domain A can also be separated, which contradicts the assumption that A is connected. Therefore, f(A) must be connected.
These results highlight that properties like compactness, path-connectedness, and connectedness are preserved under continuous functions, while other properties, such as being open or closed, are not necessarily preserved. Note here these results are proved in a general metric space setting.
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